The generalized Satake diagram classification conjecture for reflection-equation solutions

Let $\mathfrak g$ be a semisimple complex Lie algebra, let $V$ be its vector representation, and let

$$\rho:U_q(\mathfrak g)\to\operatorname{End}(V).$$

For the associated matrices $R$ and $R^\phi$, a matrix $K\in\operatorname{GL}(V)$ is a solution of the matrix reflection equation when it satisfies

$$R_{21}(\operatorname{Id}\otimes K)R^\phi(K\otimes\operatorname{Id})=(K\otimes\operatorname{Id})(R^\phi)_{21}(\operatorname{Id}\otimes K)R.$$

A generalized Satake diagram $(X,\tau)$ determines a coideal subalgebra $B(X,\tau)$ and universal K-matrix $\mathcal K(X,\tau)$.

Generalized Satake diagram classification conjecture.

1. If $K\in\operatorname{GL}(V)$ is a solution of the matrix reflection equation, then there exists a generalized Satake diagram $(X,\tau)$ such that $K$ is proportional to $\rho(\mathcal K(X,\tau))$.
2. The only quasitriangular coideal subalgebras of $U_q(\mathfrak g)$ are the pairs $(B(X,\tau),\mathcal K(X,\tau))$ associated with generalized Satake diagrams.

The first assertion connects all vector-representation solutions of the reflection equation with universal K-matrices, while the second gives the corresponding classification of quasitriangular coideal subalgebras. The paper reports verification in several low-dimensional and classical cases, but the general classification remains open.